91Ó°ÊÓ

Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. $$\left(t^{2}+1\right)^{3} y y^{\prime}(t)=t\left(y^{2}+4\right)$$

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. $$y^{\prime}(t)=y-3, y(0)=1$$

Solve the following initial value problems. $$y^{\prime}(t)-2 y=8, y(0)=0$$

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. $$y^{\prime}(t)=4-y, y(0)=-1$$

Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume \(C, C_{1}, C_{2}\) and \(C_{3}\) are arbitrary constants. $$g(x)=C_{1} e^{-2 x}+C_{2} x e^{-2 x}+2 ; g^{\prime \prime}(x)+4 g^{\prime}(x)+4 g(x)=8$$

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. $$y^{\prime}(t)=y(2-y), y(0)=1$$

Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume \(C, C_{1}, C_{2}\) and \(C_{3}\) are arbitrary constants. $$u(t)=C_{1} t^{2}+C_{2} t^{3} ; t^{2} u^{\prime \prime}(t)-4 t u^{\prime}(t)+6 u(t)=0$$

Solve the following initial value problems. $$u^{\prime}(x)=2 u+6, u(1)=6$$

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. $$y^{\prime}(x)=\sin x, y(-2)=2$$

Solve the initial value problem and graph the solution. Let \(r\) be the natural growth rate, \(K\) the carrying capacity, and \(P_{\mathrm{o}}\) the initial population. $$r=0.2, K=300, P_{0}=50$$